Theorem infcntss 8092

Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)

Hypothesis

Ref	Expression
infcntss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
infcntss	⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Distinct variable group:   𝑥,𝐴

Proof of Theorem infcntss

Step	Hyp	Ref	Expression
1		infcntss.1	. . 3 ⊢ 𝐴 ∈ V
2	1	domen 7827	. 2 ⊢ (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 7864	. . . . 5 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω)
4	3	anim2i 590	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ ω ≈ 𝑥) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
5	4	ancoms 467	. . 3 ⊢ ((ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
6	5	eximi 1750	. 2 ⊢ (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
7	2, 6	sylbi 205	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1694   ∈ wcel 1975  Vcvv 3168   ⊆ wss 3535   class class class wbr 4573  ωcom 6930   ≈ cen 7811   ≼ cdom 7812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-er 7602  df-en 7815  df-dom 7816

This theorem is referenced by: (None)